If I read you aright, your answer is a qualified Yes, that moral thought is natural to man, sourced in human nature.
The reason I (dunno about Hume) can't give a straight yes/no answer is this - the term 'moral intuitions' lends a moral framing which didn't exist when these intuitions evolved. Simple as that. In themselves they are factually described as evolved responses which enable our particular species' survival. And from a Selfish Gene pov, the survival of the body until reproduction where adaptation can occur, is the way genes survive from generation to generation. This is the Source, which doesn't require or imply imo, the prefix ''moral''. (Your opinion might differ!)
I can't give you a precise answer, but we now have the outline of a compelling account coming together of how these survival adaptations came to be conceived of as moral, about right and wrong for its own sake.The qualification has to do with a distinction you draw between moral intuition on the one hand and moral thought (intuition framed by discourse) on the other. Have I read you correctly?
If so, what are these moral intuitions before they are "conditioned" by discursive framing? And whence this discursive moral framing? How do we come by this template?
The pro-social /unselfish predispositions are what the term 'moral intuitions' usually refer to. These are neurological adaptations primarily enabling social bonding (in turn rooted in the evolution into mammals who have to care for their helpless offspring - remember Pat Churchland? ) Things like love, caring, cooperation, fairness, loyalty, trust, altruism, etc.
Along with that came reputation (being known as a trustworthy cooperator), private guilt and public shame. And despising and punishing selfish cheaters who disrupt group cohesion This is the source of the approval/disapproval response.
And suspicion of strangers, 'the other', as potential competitors and enemies (The tribal response - tribalism evolved when we lived in tribes, natch, now we don't it's a big problem).
It's not hard to imagine how these came to take on connotations of right and wrong. Religious type beliefs shaping the concept of morality and placing it in the world (or heavens) outside of us, with rules codifying Oughts. Then cultural transmission via religion, leaders, institutions, education, archetypal myths and stories, social mores and so on, reinforcing and developing them.
Natural philosophy, and what philosophy has come to be, playing a role too in conceptualising morality as thing in itself, in the world, discoverable through religious revelation, or reason or whatever. And we're dropping in on Hume as he's saying ''Hang on a minute, logic and reasoning can't find this morality everybody talks about somewhere out there in the world, it's just us going yuck or yummy!''